[/quote]
,
,
is the midpoint of 
,
the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
.
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
be the set of positive real numbers. Find all functions 
such that, for all 
,
does the number 
leave a remainder of 
when divided by 
[/quote]
be a trapezoid with 
parallel to 
,
,
,
.
and 
on segment 
,
[/quote]
of positive integers less than or equal to 20 are there such that 
,
,
are all integers?
Y by Pleaseletmewin, CELLSecret, channing421, jacoporizzo, Random_Person_921, AT2005, SuperJJ, samrocksnature, ab456, megarnie, mathtiger6, M_2007, Lamboreghini, jmiao, aidan0626
Hello AoPS!
STEAM For All is excited to announce its third virtual
The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by volunteers with years of experience in competition math! Students may only take one exam. Like the AMCs, students in and below 10th grade can take the OMC 10 while students in and below 12th grade can take the OMC 12. Students will have a great opportunity to challenge themselves with new questions, listen to experienced competitors explain the most difficult questions, and be recognized in an awards ceremony!
Register here! Registration ends Friday, October 28, 2022, at 11:59 PDT.
The competition will be held on Saturday, October 29th, 2022 from 2 - 4:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
2:00 - 2:15 PM: Introduction
2:15 - 3:30 PM: OMC 10/12 Competition
3:30 - 4:15 PM: Problem Review Session
4:15 - 4:30 PM: Awards and Closing Ceremony
For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
More Questions? Contact us at ocmathcircle@gmail.com or ask in this thread!
We hope that you join us for the upcoming competition!
STEAM For All is excited to announce its third virtual
OMC 10/12!
The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by volunteers with years of experience in competition math! Students may only take one exam. Like the AMCs, students in and below 10th grade can take the OMC 10 while students in and below 12th grade can take the OMC 12. Students will have a great opportunity to challenge themselves with new questions, listen to experienced competitors explain the most difficult questions, and be recognized in an awards ceremony!
Register here! Registration ends Friday, October 28, 2022, at 11:59 PDT.
The competition will be held on Saturday, October 29th, 2022 from 2 - 4:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
2:00 - 2:15 PM: Introduction
2:15 - 3:30 PM: OMC 10/12 Competition
3:30 - 4:15 PM: Problem Review Session
4:15 - 4:30 PM: Awards and Closing Ceremony
For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
More Questions? Contact us at ocmathcircle@gmail.com or ask in this thread!
We hope that you join us for the upcoming competition!
works. But then you have the sneakier cases to work with, and that's where I think things can get messy and easy to miss stuff (reminiscent of 2018 AIME I #9).
,
,
,
,
,
Did anyone actually get their scores back?
